Question

How can i calculate p(x,y)? (details inside)

between two electric poles, there are 2 electricity cables - one on top of the other.
the distribution of the number of birds who sitt on top of each of the cables is binomic with the parametrs 30 and 0.5 )there's no dependency between the number of birds on each of the cables

X - the number of birds on the lower cable
Y - the number of birds on both cables together

what is p(x,y)?

Answer 1

`p(x,y) = (C(30,y-x) * C(30,x)) / 2^60`

Explanation:

`"Let "X_1" and "X_2" be independent binomial random variables"`
`"where "X_i" has a Binomial("n_i, p") distribution for i = 1, 2. "`
`"Then "X_1 + X_2" has a binomial distribution with "n_1+n_2` `"trials and probability of success p."`

`"So Y has a binomial distribution with parameters 60 and 0.5."`

`"But we do not need this fact. We simply have :"`

`p(x,y) = p(y|x) p(x) = ((C(30,y-x))/2^30)*((C(30,x))/2^30)`

`= (C(30,y-x) * C(30,x)) / 2^60`

`(z = y - x = "the number of birds on the upper cable,"`
`"so z follows the Binomial(30, 0.5) distribution)."`